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Number Theory Days 2011 Schedule

Abstract: (joint work with Stefan Müller-Stach) Periods are numbers that one gets by integrating a rational differential form over a cycle. They form a very interesting subalgebra of the complex number, including e.g. \pi, \log(2), \zeta(n) for all n\in \Z The period conjecture says that the only relation between these periods are the obvious ones. This is a very strong assertion on transcendence. In the talk I am going to make this statement precise by introducing Kontsevich's algebra of formal periods. Following ideas of Kontsevich and Nori, we then show that the corresponding proalgebraic scheme is a torsor under the motivic Galois group in the sense of Nori.

15.15-15.45: Pause

15.45-16-45: Pierre Parent Rational points on X_0^+ (p^r )

(Joint work with Yu. Bilu and M. Rebolledo). The points of the modular curves X_0^+ (p^r ) over a field K, for {r>1} and p a prime number, parametrize quadratic K-curves f degree $p^r$ or, when ${r=2}$, elliptic curves over $K$ endowed with a mod $p$ Galois structure which is the normalizer of a split Cartan group. The generic non-existence of such objects over number fields is part of an old conjecture of Serre regarding uniform surjectivity of Galois representations associated with elliptic curves. We show how a combination of analytic estimates on modular units, geometro-algebraic techniques about integrality, and isogeny bounds (in the recent version due to Gaudron and R\'emond) allow to obtain the triviality of {X_0^+ (p^r )({\bf Q} )} for all r>1 and all primes larger than 2.10^{11}. We then prove, with the help of computer calculations, that the same holds true for $p$ in the range {11\leq p\leq 10^{14}}, {p\neq 13}. Together with known results about very small primes, this completely solves our problem over Q... with exactly one exception in level 13.

Abstract: We discuss statistic properties of automorphisms of nilmanifolds and shows their iterations generate sequences with good independence properties. It turns out that this question is closely related to the problem of Diophantine approximation of algebraic numbers. This is a joint work with R. Spatzier.

Abstract: Given an algebraically closed complete valued field of characteristic p, we construct a curve over Qp and classify vector bundles on it. To some objects in p-adic Hodge theory we associate Galois equivariant vector bundles on this curve. As a particular case of the classification of vector bundles on this curve we find back the two main theorems of p-adic Hodge theory: weakly admissible is equivalent to admissible and De Rham implies potentially semi-stable. This is joint work with Jean-Marc Fontaine.

In the 1980's, G. Wuestholz proved a deep and far reaching theorem, known as the "Analytic Subgroup Theorem", which continues to have an extensive impact on transcendental number theory. We recall this theorem, and its consequences for transcendence of special values of modular and hypergeometric functions, in the context of variations of polarized Hodge structures of level 1 (Shimura varieties). These results are related to conjectures of Andre-Oort and Pink. We then describe some hopes for similar results in the context of variations of Hodge structures of higher level.